A topological group (in particular, a Lie group) for which the underlying topological space is simply-connected. The significance of simply-connected groups in the theory of Lie groups is explained by the following theorems.
1) Every connected Lie group is isomorphic to the quotient group of a certain simply-connected group (called the universal covering of ) by a discrete central subgroup isomorphic to .
2) Two simply-connected Lie groups are isomorphic if and only if their Lie algebras are isomorphic; furthermore, every homomorphism of the Lie algebra of a simply-connected group into the Lie algebra of an arbitrary Lie group is the derivation of a (uniquely defined) homomorphism of into .The centre of a simply-connected semi-simple compact or complex Lie group is finite. It is given in the following table for the various kinds of simple Lie groups.'
In the theory of algebraic groups (cf. Algebraic group), a simply-connected group is a connected algebraic group not admitting any non-trivial isogeny , where is also a connected algebraic group. For semi-simple algebraic groups over the field of complex numbers this definition is equivalent to that given above.
|[a1]||G. Hochschild, "The structure of Lie groups" , Holden-Day (1965)|
|[a2]||R. Hermann, "Lie groups for physicists" , Benjamin (1966)|
|[a3]||J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1|
Simply-connected group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Simply-connected_group&oldid=18749